Question

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x4+x3-x2+3; no real zero less than -2

Accepted Answer

First, understand the problem: we need to show that any real zero $$ x  of $$the polynomial function $$ f(x) = x^4 + x^3 - x^2 + 3  is $$not less than $$ -2 . In $$other words, if $$ f(x) = 0 $$, then $$ x \geq -2 . $$Evaluate the polynomial at $$ x = -2  to $$check the value of $$ f(-2) . $$This helps determine if $$ -2  is a $$root or if the function changes sign around this point. Write $$ f(-2) = (-2)^4 + (-2)^3 - (-2)^2 + 3 . $$Analyze the behavior of $$ f(x) $$ for values less than $$ -2 . $$For example, pick a test value such as $$ x = -3 $$ and compute $$ f(-3)  to $$see if the function is positive or negative there. This helps identify if the function crosses the x-axis to the left of $$ -2 . $$Use the Intermediate Value Theorem: if $$ f(x) $$ does not change sign between $$ -10 $$ and $$ -2  (or $$any interval to the left of $$ -2 $$), then there are no real zeros less than $$ -2 . $$Optionally, analyze the derivative $$ f'(x)  to $$understand the shape and monotonicity of the polynomial on the interval $$ (-1	ext{, } -2) $$, which can support the conclusion about the location of zeros.

Show that the real zeros of each polynomial function satisfy the given conditions. ƒ(x)=x⁴+x³-x²+3; no real zero less than -2

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Polynomial Functions and Their Zeros

Evaluating Polynomial Values at Specific Points

Using Inequalities to Bound Zeros